Zero Acceleration

1. The problem statement, all variables and given/known data
Let c be a path in R^3 with zero acceleration. Prove that c is a straight line or a point.

2. Relevant equationsF(c(t)) = ma(t)a(t) = c''(t)

3. The attempt at a solution
so i know that since the acceleration is zero, the velocity must be constant, and when you integrate a constant, you get a straight line...but how to I prove mathematically that the velocity is constant, because you can't integrate 0dt, as far as I know?

1. The problem statement, all variables and given/known data
Let c be a path in R^3 with zero acceleration. Prove that c is a straight line or a point.

2. Relevant equationsF(c(t)) = ma(t)a(t) = c''(t)

3. The attempt at a solution
so i know that since the acceleration is zero, the velocity must be constant, and when you integrate a constant, you get a straight line...but how to I prove mathematically that the velocity is constant, because you can't integrate 0dt, as far as I know?

Damn, I hate mixed "physics" and "mathematics" problems! You or whoever set this problem, should know that a "path" DOES NOT HAVE an "acceleration". I expect this problem should be "find the equation of motion of a particle whose trajectory is a given path in R3 with acceleration 0. Show that the path is either a straight line or a point". Then you would begin with [itex]\vec{a}= d\vec{v}/dt=[/itex] and go from there.

The easiest way would have been to recognise that acceleration is a vector quantity, it is affected both by direction or magnitude. No acceleration, no change in direction, which means constant gradient. Simple as that.